**Example 1 (Asymptotes)**

*Let*

*What is the domain of ? What are the roots and -intercepts of ? Find the horizontal and vertical asymptotes of .*

* Solution *The

**domain**of is the set on which is defined. As a quotient of continuous functions, is defined when the denominator is not zero:

That is the domain of is (*all the real numbers except -1 and 2. Note now that*

* *).

The **roots** of a quotient of continuous functions occur at points in the domain when the numerator is zero:

That is has no (real) roots ( is not a real number).

The **-intercept **of is the point where cuts the -axis (*i.e. when *):

The **horizontal asymptotes **of are the behaviours of as :

.

Hence is the horizontal asymptote.

The **vertical asymptotes **of are the finite where gets very big (*has a singularity*). Following our procedure:

*Step 1:* A quotient of continuous functions can only have a vertical asymptote when the denominator is zero; i.e. (*it is *necessary but not sufficient *that the denominator is zero – hence Step 2*).

*Step 2*: Evaluate the limits

and

*Step 3*: As these limits evaluate to infinity there are vertical asymptotes at and .

**Comment **To carry out the full analysis of this function to sketch it is quite messy – we need a computer to find the split points of and .

**Example 2 (all elements considered)**

*Let *

*Find the domain of . Find the roots and -intersections of . Find the horizontal and vertical asymptotes of . Find the intervals where is increasing/ decreasing. Find the locations of the local extrema (*local max/min*). Find where the function is concave up/ concave down. Hence sketch the graph.*

** Solution **The

**domain**of is where

Domain of is .

The **roots **of are where

.

The **-intercept**, .

The **horizontal asymptotes**

.

Hence the horizontal asymptote . *(Which is a line of slope and -intercept . (Cf. ))*.

The **vertical asymptotes:**

*Step 1: **.*

*Step 2: *

*Step 3: *Hence is a vertical asymptote.

To find where is **increasing/ decreasing**, we have to find the split points of (*the only place can change sign; i.e. the only place where can go from increasing to decreasing or vice versa . Remember increasing; decreasing. The split points are where or undefined*). By the quotient rule, is undefined when the denominator is zero; i.e. . By the quotient rule:

The other split points are where , and in this case this occurs when ; i.e. . Now *we test in between the split points *to see if is positive/negative ( increasing/decreasing). The split points are so the intervals to test are:

Choose test points :

Hence on and on ; i.e. increasing on and decreasing on .

To find out the locations of the** local extrema**, we do the following schematic of (*If is increasing on an interval draw it increasing, if is decreasing on an interval draw it decreasing. Beware of singularities! At these points is not continuous* (like it is at and )).:

*(please ignore the vertical line joining them up) *It is clear from this schematic that there is a local maxima at and a local minima at . These have coordinate and .

To find where is **concave up/ concave down** we find the split points of . By the quotient rule, as

,

is undefined at . By the quotient rule:

Because we have already considered the case when , we may cancel the (*i.e. dividing by is not troublesome as we have already identified as a split point and hence we will not be dividing by zero*):

Hence the only split point of is . Now *we test in between the split points *to see if is positive/negative ( concave up/ down). The split point is so the intervals to test are:

Choose test points :

Hence on and on ; i.e. concave up on on and concave down on .

To **sketch the graph **take all these points into consideration (*click for a sharper image*):

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